%% Generalization of 19.
\item 
Instead of the 'natural' and 'clamped' boundary conditions,
the matlab built-in 'spline' function enforces the so-called
'not-a-knot' boundary condition by requiring 
$S'''(x)$ to be continuous across $x_1$ and $x_{n-1}$
(although you could impose the clamped boundary condition as well. 
See 'help spline' for details).

Show that this is equivalent to
assigning one cubic polynomial on each of the
intervals $[x_0, x_2], [x_2, x_3], [x_3, x_4], \cdots,
[x_{n-4}, x_{n-3}],
[x_{n-3}, x_{n-2}],
[x_{n-2}, x_{n}]$, with the usual continuity conditions
across the 'knots' 
$x_2, x_3, \cdots, x_{n-2}$ 
and two extra conditions 
$S(x_1)= f(x_1)$,
$S(x_{n-1})= f(x_{n-1})$.
In this sense, 
neither $x_1$ nor $x_{n-1}$ is a knot, hence the name.

\item Interpolate the function $f(x) = {1 \over 1+x^2}$
on $[-5,5]$ with equally spaced nodes \\
$-5, -4.5, -4, \cdots, 4.5, 5$ using
matlab built-in spline, the Lagrange polynomials
and Hermite polynomials, respectively. 
Plot the interpolated polynomials on $-5:0.05:5$.

Hand in your source code by Sun, Nov 28 via email to the
homework account.
Put all your codes in a single file and name it
as na10f\_hw10\_your-id-number.m.
Execute this file again before handing in.
Sample files 
will be provided on the course homepage to explain
how to plot more than one figures at the same time. 

% \item Hermite polynomial
% Section 3.4: Problems 25.

\item
% Section 4.1: Problems 7(a), 19, 22, 24, 28, 29.
Section 4.1 is put in homework 11.

%% \item 
%% In continuation of problems 24 and 19 in section 4.1,
% Use the sample file.
% What you will observe is known as Runge's phenomenum.
% This is the reason to use piecewise polymomial interpolaton,
% instead of a high order polynomial interpolation.
\end{enumerate}
\end{document}